Angular Displacement Calculator

Formula

How this is calculated

Angular displacement θ is the angle through which a body rotates, measured in radians (rad). One full revolution is 2π rad = 360°. Unlike linear displacement, it can accumulate beyond 2π for multiple rotations.

Three methods yield θ depending on your data. If you know the arc length s and the radius r of the circular path, θ = s / r (definition of radian measure). This is exact for any arc on a circle of known radius — 3.14 m on a 1 m circle sweeps out almost exactly π radians. If you know the initial angular velocity ωᵢ, the (constant) angular acceleration α, and the time t, the kinematic equation θ = ωᵢ·t + ½·α·t² applies, analogous to linear kinematics (s = u·t + ½·a·t²). For uniformly accelerated rotation you can also use θ = ½·(ωᵢ + ωf)·t, which needs only the initial and final angular velocities and the time.

All three methods assume rigid-body rotation about a fixed axis and constant angular acceleration. Results are converted to degrees (×180/π) and revolutions (÷2π) for convenience. Negative values of θ mean clockwise rotation (with counterclockwise defined as positive). A zero radius or zero time in the relevant fields returns no result.